Class 17: Structural Induction

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Class 17: Structural Induction cs2102: Discrete Mathematics | F16 uvacs2102.github.io David Evans University of Virginia 0

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Plan Recursive Data Types: Lists Recap List Operations Proving Properties about All Lists Structural Induction Trees 1

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Recap: Lists null: prepend: × ⟶ first: List ⟶ rest: ⟶ empty: ⟶ Definition. A list is an ordered sequence of objects. A list is either the empty list (), or the result of prepend(, ) for some object and list . List Operations Constructors Observers 2

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Defining List Operations null: prepend: × ⟶ first: List ⟶ rest: ⟶ empty: ⟶ first prepend(, ) ⟶ rest prepend(, ) ⟶ empty prepend(, ) ⟶ empty null ⟶ 3

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Length of a List 4

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Length of a List Definition. The length of a list, , is: 0 if is null 1 + length otherwise = prepend , 5

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Prove: for all lists, , list_length() returns the length of the list . Definition. The length of a list, , is: 0 if is null 1 + length otherwise = prepend , 6

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Prove: for all lists, , list_length() returns the length of the list . Definition. The length of a list, , is: 0 if is null 1 + length otherwise = prepend , 7

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Concatenation Definition. The concatenation of two lists, = U , V , … , X and = U , V , … , Y is U , V , … , X , U , V , … , Y . How can we define this constructively? 8

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Concatenation Definition. The concatenation of two lists, = U , V , … , X and = U , V , … , Y is U , V , … , X , U , V , … , Y . How can we define this constructively? Also: poll on “slack breaks” Any questions about PS6, definitions so far, recursive data, etc. 9

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Concatenation Definition. The concatenation ( + ) of two lists, and , is defined as: Base case: = (empty list) + = Constructor case: = prepend(, ) for some list , object + = prepend(, + ) 12

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Length of Concatenation Prove. For any two lists, and , length( + ) = length() + length() 13

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Prove. For any two lists, and , length( + ) = length() + length() Base case: = (empty list) + = Constructor case: = prepend(, ) + = prepend(, + ) 14

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Structural Induction To prove for all objects of a data type: 1. Prove for all base objects . 2. Prove for all data type objects : ⇒ for all constructable from . 15

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Structural Induction (Data Types) Invariant Principle (State Machines) (Regular) Induction (Natural Numbers) To prove ^ prove a base case prove an inductive step quod erat demonstrandum. 16

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Structural Induction (Data Types) Invariant Principle (State Machines) (Regular) Induction (Natural Numbers) for all data type objects for all reachable states To prove ^ for all natural numbers prove a base case 0 _ base object prove an inductive step ⇒ ( + 1) ⇒ for all constructable from ⇒ for all reachable from quod erat demonstrandum. 17

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Trick-or-Treat Protocols 18

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“Trick or Treat” 19 Tricker initiates the protocol by making a threat and demanding tribute Victim either pays tribute (usually in the form of sugary snack) or risks being tricked

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“Trick or Treat” 20 Tricker initiates the protocol by making a threat and demanding tribute Victim either pays tribute (usually in the form of sugary snack) or risks being tricked Tricker must convince Victim that she poses a credible threat: prove she is a qualified tricker

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Trick-or-Treat Trickers? Victim 21 Any problems with this?

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Proof without Disclosure How can the tricker prove their trickability, without allowing the victim to now impersonate a tricker? 22

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Challenge-Response Protocol 23 Prover: proves knowledge of by revealing (, ) . Verifier: convinced prover knows , but learns nothing useful about . Verifier: picks random . Need a one-way function: hard to invert, but easy to compute.

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Example: RSA 24 Ee (M ) = Me mod n Dd (C ) = Cd mod n Correctness property: Ee (Dd ()) =

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Trick-or-Treat Trickers? Victim 25

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Trick-or-Treat Trickers? Victim 26 How does victim know e and n? Verify: n = nmod =

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27 “Elsa #253224”, = 3482..., = 1234... signed by Tricker’s Buroo Verify: n = nmod = Verify Tricker’s Buroo signature on certificate

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28 “virginia.edi”, = 3482..., = 1234... signed by Certificate Authority Verify and Decrypt: p n () = Verify signature on certificate Server

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Binary Trees A binary tree is either: - null or - node: (Tree, Object, Tree) 30

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Tree Operations A binary tree is either: - null or - node: (Tree, Object, Tree) 31

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A binary tree is either: - null or - node: (Tree, Object, Tree) null: node: × × → label: → left: → right: → empty: → Tree Operations 32

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Structural Induction To prove for all objects of a data type: 1. Prove for all base objects . 2. Prove for all data type objects : ⇒ for all constructable from . Does this work for trees? 33

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Structural Induction To prove for all objects of a data type: 1. Prove for all base objects . 2. Prove for all data type objects : ⇒ for all constructable from . Tree Constructors null: node: × × → 34

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Binary Structural Induction To prove for all objects of a data type: 1. Prove for all base objects . 2. Prove for all data type objects U , V : U ∧ V ⇒ for all constructable from U and V. 35

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Number of Labels Prove: The number of labels in a binary tree with nodes is . null: node: × × → 36

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Charge Enjoy Halloween Don’t be victimized by any unsubstantiated threats! Problem Set 7 due Friday 37